Accuracy Analysis and Synthesis of Planar Mechanism for Antenna Based on Screw Theory and Geometric Coordination

Abstract

To address the deployment accuracy issues of multi-frequency band reflector antennas, this study takes a hexagonal prism modular deployable antenna as an example and proposes an accuracy design method. This paper proposes a screw-theory-based sub-chain precision analysis method. This method constructs a virtual screw model of rod length errors and hinge gap errors. Based on geometric relationships, a multi-loop point position error model is established, and accuracy surfaces considering rod length errors and hinge gap are output using MATLAB R2024b. By outputting the relationship curves of single-rod errors relative to point errors, the linearized influence law of individual rods on precision is further elucidated. Simulation results demonstrate the reliability of the error modeling theory. Based on the established cost-effective precision model and the minimum point error, which is obtained by using the numerical iterative method, the optimal solution for error parameters is obtained.

1. Introduction

In modern radar, communication, and remote sensing technologies, reflector antennas serve as one of the core components, with their performance directly influencing the sensitivity, selectivity, and anti-interference capability of the system. Due to their complex geometric structures and multi-band sharing features, multi-frequency band reflector antennas have become a research hotspot in recent years [1]. However, the deployment accuracy of such antennas has long been a critical factor limiting their practical applications. With the rise of modular deployable antenna technology, their potential in radar, communication, and remote sensing applications is gradually being recognized, but achieving a high-accuracy design for modular deployable antennas remains an urgent challenge [2]. Precise antenna deployment requires not only highly consistent geometric positioning and orientation of reflector elements but also involves multi-faceted co-optimization of material properties, structural stiffness, and antenna-matching networks [3]. Therefore, improving the deployment accuracy of modular deployable antennas not only requires breakthroughs in theoretical research but also the development of efficient and reliable numerical simulation and optimization methods. This study focuses on the accuracy design of modular deployable antennas, exploring novel ideas and methods to address the deployment accuracy challenges of multi-frequency band reflector antennas.

The deployment accuracy of modular deployable antennas is influenced by various factors, including machining accuracy, assembly accuracy, structural deformation, and component wear. Among these, rod length error and hinge gap error are the primary factors affecting the positional accuracy of the deployment mechanism [4]. The accuracy design of modular deployable antennas is a crucial approach for enhancing deployment accuracy, with accuracy analysis and accuracy synthesis being its main components. Accuracy analysis aims to reveal the impact of various errors on the end accuracy. While accuracy synthesis is to formulate or reasonably allocate various error parameters according to the unfolding accuracy requirements of a specified mechanism, which is the inverse problem of mechanism accuracy analysis [5]. At present, several research institutions and scholars worldwide have conducted in-depth research on this topic.

In terms of the accuracy analysis of mechanisms, Ding et al. [6] proposed an accuracy analysis method based on matrix block modeling, decomposing complex deployment models into several single-loop mechanism computational models to separately evaluate the impact of hinge gaps on accuracy. Liu et al. [7] introduced a direct linearization method, expressing component tolerances, hinge gaps, and deformation errors as linear vectors to establish a vector-loop model for mechanism closure, and calculated the positional deviation of the closed vector loop through homogeneous transformation matrices. Dong et al. [8] proposed a modeling approach for mechanisms with hinge gaps based on massless rod models; this method employed the D-H method [9] to construct geometric relationship models for three closed loops of an antenna and integrated the massless rod model into the mechanism’s error model. Chen et al. [4] developed an accuracy analysis method that simultaneously accounts for hinge gaps and rod length error, by applying screw theory; they equated hinge gap and rod length error to virtual screws, established a comprehensive error model, and calculated the positional deviation at the open-chain output point using homogeneous transformation matrices. Ding et al. [10] employed the vector differential method to create an error model for a six-degree-of-freedom parallel mechanism considering hinge gaps; they also proposed a pose accuracy analysis method based on Sobol sequence quasi-Monte Carlo simulation. Hafezipour et al. [11] examined the effects of link dimension deviations and hinge gaps on the positional accuracy of spatial mechanisms and robotic end effector, proposed an error modeling formulation based on the direct linearization method. Some studies [12,13,14,15] conducted geometric error modeling and sensitivity analysis for parallel robots, identifying the primary factors influencing the accuracy of parallel mechanisms; Liu et al. [16,17] established an error model for a 4/2-type Stewart mechanism that considers bearing clearance and dimensional errors in the driving branches, based on the error model; they conducted a sensitivity analysis of various error components of the mechanism. In terms of the accuracy synthesis of mechanisms, Ni et al. [18] designed a fully rotational parallel robot and modeled its errors, then conducted parameter tolerance design using a minimum-cost model. Liu et al. [19] performed accuracy synthesis for the TriMule hybrid mechanism, with the objective function aimed at minimizing manufacturing costs and accuracy requirements as constraints. Han et al. [20] established configuration optimization design criteria based on a sensitivity analysis of various parameters affecting the end accuracy of parallel mechanisms, enabling a multi-objective accuracy optimization design for the main structural parameters of parallel mechanisms. Gao et al. [5] employed genetic algorithms for the optimized allocation of parameter tolerances for a 7-DOF robotic arm, with error minimization as the objective function and manufacturing costs as constraints. Yao et al. [21] constructed an accuracy synthesis model to minimize manufacturing costs under accuracy constraints and solved the optimization problem using genetic algorithms. Wu et al. [22] established an error model for a 3-DOF parallel robot using the closed-loop vector method, conducting sensitivity analysis via the Sobol sequence-based quasi-Monte Carlo (Sobol-QMC) method, to formulate an accuracy synthesis objective function targeting minimal manufacturing costs, and determine reasonable tolerance ranges for various errors using genetic algorithms.

Currently, most domestic and foreign scholars use screw theory or differential methods for precision analysis, but there are many limitations. For example, the differential method requires establishing a multi-single error model first when solving precision problems for complex models, and then combining the mechanism’s position equation for differential solving, making the process extremely complex. Screw theory expresses errors using screws and represents the posture of components through coordinate homogeneous transformations, but when dealing with multi-loop coupled models, the coordination relationships between loops are not considered, which will lead to imprecise error analysis. Based on geometric relationships, this paper establishes a multi-loop error model, converts the deformation coordination of coupled loops into a nonlinear equation group, and solves it by using numerical solution methods.

2. Antenna Structure

The modular deployable antenna exhibits strong topological performance, high stability, and superior deployment accuracy. By adjusting the quantity, type, arrangement, and combination of modules, it enables rapid scaling of deployment dimensions, making it an ideal structural solution for meeting the multi-band operational requirements of multi-frequency band reflector antennas [1]. Modular deployable antennas typically consist of a central module and multiple peripheral modules. The overall mechanism not only features synchronized folding and deployment capabilities but also achieves multi-band compatibility through variable stiffness design, as illustrated in Figure 1.

The basic rib unit structure is shown in Figure 2. When the rib unit is folded, the slider is positioned at the bottom of the central beam. During deployment, the slider moves upward while the support rod drives the other rods to unfold. When the axes of the small and large diagonal bracing rods are fully aligned, the entire mechanism is locked, and the rib unit becomes a stable structure. The dimensions of each rod are listed in

Table 1. Rod length of the basic unit.

Name	${\mathit{l}}_1$	${\mathit{l}}_2$	${\mathit{l}}_3$	${\mathit{l}}_4$	${\mathit{l}}_5$	${\mathit{l}}_6$	${\mathit{l}}_7$
Size (mm)	750	190	610	750	164	190	95

. Due to the symmetric coupling characteristics of the basic rib units, the deployment accuracy of the entire configuration can be reflected by the mechanism of these basic rib units.

3. Error Modeling and Accuracy Analysis

Accuracy analysis is a crucial aspect of mechanism design. To investigate the influence of rod length error and hinge gap on the accuracy of deployable mechanisms, a multi-closed-loop unit accuracy modeling approach is proposed. Based on screw theory, the rod length error and hinge gap error are expressed parametrically. The error model of each branch is established. A comprehensive error model is then established by using geometrical compatibility that incorporate these error parameters.

3.1. Error Modeling Approach

The screw method is an accuracy analysis approach based on screw theory. Its fundamental concept involves representing the position of components using screws. It solves and analyzes position error through coordinate transformation techniques [23].

Based on the structure of the single-degree-of-freedom deployable unit, the central beam is regarded as the base, and its error is neglected. It is assumed that slider 7 and support rod 8 have precise accuracy, and its error influence is not considered. The vector of errors in the mechanism’s end can be described as the difference between the actual and nominal position vectors of the end point of each kinematic chain, and then obtaining the vector of errors in the mechanism’s end. The main steps are detailed as follows:

• The multi-loop mechanism is decomposed into multiple serial sub-chains. The model without errors for each sub-chain is established using screw transformation matrices.
• The gap and rod length errors are parameterized using virtual screws. Subsequently, the comprehensive error model of each sub-chain is established by applying screw transformation matrices again.
• The vector of errors in each sub-chain is obtained by subtracting the terminal position vectors of the same chain from the comprehensive error model and the model without errors.
• The vector of errors in a single chain is calculated by establishing the geometrical compatibility model. The magnitude of the vector is calculated, and the accuracy surface is visualized using MATLAB R2024b.

3.2. The Model Without Errors

The planar multi-closed-loop mechanism is the focus of this analysis. A simplified version of Figure 2 from Section 2 is presented as Figure 3. For the planar antenna, the positions of the four vertices of the support panel determine the unfolding accuracy of the antenna panel, making these vertex coordinates the key points for accuracy analysis. Considering the central AB rod as the base and neglecting its error effects, the BC rod serves as the working rod subjected to external loads. Based on the principle of error accumulation and transmission, this section primarily analyzes the accuracy at point D.

Divide A–B–C–D–E into three kinematic chains, namely the B–C–${\mathrm{D}}_1$ chain, the B–E–${\mathrm{D}}_2$ chain, and the A–${\mathrm{D}}_3$ chain. Establish the screw models for each sub-chain, as shown in Figure 4.

A Cartesian coordinate system is established at point B. In the kinematic chain B–C–${\mathrm{D}}_1$ solving for the kinematic error at point ${\mathrm{D}}_1$. In the kinematic chain B–E–${\mathrm{D}}_2$, solving for the kinematic error at point ${\mathrm{D}}_1$. In the kinematic chain A–${\mathrm{D}}_3$, solving for the kinematic error at point ${\mathrm{D}}_3$.

In the kinematic chain B–C–${\mathrm{D}}_1$, the length of the BC rod is $l_1$ and the length of the ${\mathrm{C}\mathrm{D}}_1$ rod is $l_2$, with hinge gap rotation angles ${\theta }_1$ and ${\theta }_2$, and their corresponding screws being ${\$}_1$ and ${\$}_2$. In the kinematic chain B—E—${\mathrm{D}}_2$, the length of the BE rod is $l_5$, and the length of the ${\mathrm{E}\mathrm{D}}_2$ rod is $l_3$, with hinge gap rotation angles ${\theta }_5$ and ${\theta }_3$, and their corresponding screws being ${\$}_5$ and ${\$}_3$. In the kinematic chain A–${\mathrm{D}}_3$, the length of the ${\mathrm{A}\mathrm{D}}_3$ rod is $l_4$, with a hinge gap rotation angle ${\theta }_4$, and its corresponding screw being ${\$}_4$.

The screw parameters of the closed-loop unit mechanism are listed in

Table 2. List of screw parameters.

Screw	${\mathit{S}}_{\mathit{i}}$	${\mathit{S}}_{\mathit{o}\mathit{i}}$	${\mathit{\theta }}_{\mathit{i}}$	${\mathit{t}}_{\mathit{\xi }}$
${\$}_1$	(0,0,1)	(0,0,0)	${\theta }_1$	0
${\$}_2$	(0,0,1)	$(l_1$,0,0)	${\theta }_2$	0
${\$}_3$	(0,0,1)	$(l_3$,0,0)	${\theta }_3$	0
${\$}_4$	(0,0,1)	$(l_6$,0,0)	${\theta }_4$	0
${\$}_5$	(0,0,1)	(0,0,0)	${\theta }_5$	0

.

The coordinates of the kinematic chain B–C–${\mathrm{D}}_1$ are expressed as follows:

$${}^{D_1}P=T_1{T}_2{}^{D_1^0}P_0$$

(1)

The coordinates of the kinematic chain B–E–${\mathrm{D}}_2$ are expressed as follows:

$${}^{D_2}P=T_5{T}_3{}^{D_2^0}P_0$$

(2)

The coordinates of the kinematic chain A–${\mathrm{D}}_3$ are expressed as follows:

$${}^{D_3}P=T_4{}^{D_3^0}P_0$$

(3)

The above derivation process does not consider error, making it an idealized model of a planar four-bar mechanism. Therefore, Equations (1)–(3) are expressed as follows:

$$T_1{T}_2{}^{D_1^0}P_0=T_5{T}_3{}^{D_2^0}P_0=T_4{}^{D_3^0}P_0$$

(4)

The matrices and vectors in Equations (1)–(3) can be found in Appendix A.1. In Section 3, the screw rotation transformation can be combined with the translational transformation to form a $4\times 4$ homogeneous transformation matrix T describing general transformations. Its block matrix form is as follows:

$$\mathrm{T}=\left[\begin{array}{cc}\mathrm{R}& \mathrm{P}\\ 0& 1\end{array}\right]$$

(5)

In the equation, $\mathrm{R}$ represents the rotation transformation matrix that rotates θ around the z axis, as shown in Equation (6). The position vector P of the screw at this time is expressed as follows [23]:

$$\mathrm{R}={\mathrm{R}}_Z\left(\theta \right)=\left[\begin{array}{ccc}\mathrm{c}\mathrm{o}\mathrm{s}\theta & -\mathrm{s}\mathrm{i}\mathrm{n}\theta & 0\\ \mathrm{s}\mathrm{i}\mathrm{n}\theta & \mathrm{c}\mathrm{o}\mathrm{s}\theta & 0\\ 0& 0& 1\end{array}\right]$$

(6)

$$P=\left[\begin{array}{c}\mathrm{E}-\mathrm{R}\end{array}\right]S_0+t$$

(7)

For a planar six-bar linkage, the main errors affecting the kinematic accuracy are rod length error and hinge gap error. Below, based on screw theory, an error analysis model for rod length error and hinge gap is established.

3.3. Comprehensive Error Modeling

3.3.1. Screw Description of Error

The length error in the rod, which is a bilateral error, refers to the error in the center-to-center distance between two points. The rod length parameter can be expressed as $li=l\pm \Delta$, as illustrated in Figure 5a. The rod length error can be equivalently expressed as a virtual moving pair represented by the virtual screw ${{\$}^{\prime }}_{ij}(0;S_0)$.

There are many factors that affect hinge gap, including machining errors, assembly errors, and positioning errors. To comprehensively consider these error factors in the analysis, a unified error measurement standard is established. It is assumed that during the motion process, the pin always remains close to the inner wall of the hole. The hole-shaft fit (partial schematic) is shown in Figure 5b.

The eccentricity $\delta$ is the difference between the radii of the hole and the pin, which can be expressed as follows [4]:

$$\delta =r_h-r_a$$

(8)

During the hinge gap movement, any point on the shaft can contact any point on the inner wall of the hole. Therefore, the hinge gap error can be treated as a massless virtual rod with a very short length and a very limited range of rotation. Furthermore, the contact point, hole, and pin center are collinear, the virtual rod is coaxial with the rod, and the angle between the virtual rod and the rod is 0. The virtual rod is equivalent to the virtual screw ${{\$}^{\prime }}_{ih}(S;S_0)$ [2].

3.3.2. Error Modeling for Branch

The multi-loop six-bar linkage is not a fully serial mechanism, so it is crucial to propose the following assumptions before performing cumulative error modeling.

• First, the error transmission along the kinematic chain is assumed to be linear, which facilitates the linearized trend of the mechanism’s error output.
• Second, the end constraints are temporarily disregarded, as these constraints represent a complex nonlinear problem. Solving this issue could involve probabilistic methods or finite element simulation.

We bring the established models of rod length and hinge gap into the closed-loop mechanism to obtain the error model of the mechanism, as shown in Figure 6.

In Figure 6, ${\$}_1$, ${\$}_2$, ${\$}_3$, ${\$}_4$, ${\$}_5$ represent the main screws of the macro dimensions of the links; ${{\$}^{\prime }}_{12}$${{\$}^{\prime }}_{23}$${{\$}^{\prime }}_{32}$, ${{\$}^{\prime }}_{43}$, ${{\$}^{\prime }}_{53}$ are the virtual screws for rod length error; ${{\$}^{\prime }}_{1h}$, ${{\$}^{\prime }}_{2h}$, ${{\$}^{\prime }}_{3h}$, ${{\$}^{\prime }}_{4h}$, ${{\$}^{\prime }}_{5h}$ are the virtual screws for hinge gap error. The AB rod is fixed relative to the other rods, and its errors are not considered. The original closed-loop mechanism should be decomposed into open chain B–C–D₁, open-chain B–E–D₂, and open-chain A–D₃. The screw parameters for the two open chains are listed in Appendix A.3 

Table A1. List of screw parameters of comprehensive error model.

Screw	${\mathit{S}}_{\mathit{i}}$	${\mathit{S}}_{\mathit{o}\mathit{i}}$	${\mathit{\theta }}_{\mathit{i}}$	${\mathit{t}}_{\mathit{\xi }}$
${\$}_1$	(0,0,1)	$({\delta }_1$,0,0)	${\theta }_1$	0
${\$}_2$	(0,0,1)	$({\delta }_1+l_1+{\delta }_2$,0,0)	${\theta }_2$	0
${\$}_3$	(0,0,1)	$({\delta }_5+l_5+{\delta }_3$,0,0)	${\theta }_3$	0
${\$}_4$	(0,0,1)	$({\delta }_4+l_6$,0,0)	${\theta }_4$	0
${\$}_5$	(0,0,1)	$({\delta }_5$,0,0)	${\theta }_5$	0
${{\$}^{\prime }}_{12}$	(0,0,0)	(1,0,0)	0	${\Delta }_1$
${{\$}^{\prime }}_{23}$	(0,0,0)	(1,0,0)	0	${\Delta }_2$
${{\$}^{\prime }}_{32}$	(0,0,0)	(1,0,0)	0	${\Delta }_3$
${{\$}^{\prime }}_{43}$	(0,0,0)	(1,0,0)	0	${\Delta }_4$
${{\$}^{\prime }}_{53}$	(0,0,0)	(1,0,0)	0	${\Delta }_5$
${{\$}^{\prime }}_{1h}$	(0,0,1)	(0,0,0)	${{\theta }_1}^{\prime }$	0
${{\$}^{\prime }}_{2h}$	(0,0,1)	$({\delta }_1+l_1$,0,0)	${{\theta }_2}^{\prime }$	0
${{\$}^{\prime }}_{3h}$	(0,0,1)	$({\delta }_5+l_5$,0,0)	${{\theta }_3}^{\prime }$	0
${{\$}^{\prime }}_{4h}$	(0,0,1)	$(l_6$,0,0)	${{\theta }_4}^{\prime }$	0
${{\$}^{\prime }}_{5h}$	(0,0,1)	(0,0,0)	${{\theta }_5}^{\prime }$	0

.

The position of point D₁ output from the open chain B–C–D₁ is expressed as follows:

$${}^{D_1^{\prime }}P=T_{1h}^{\prime }{T}_1{T}_{12}^{\prime }{T}_{2h}^{\prime }{T}_2{T}_{23}^{\prime }{}^{D_1^0}P_0$$

(9)

The position representation of point D₂ from the open chain B–E–D₂ is expressed as follows:

$${}^{D_2^{\prime }}P=T_{5h}^{\prime }{T}_5{T}_{53}^{\prime }{T}_{3h}^{\prime }{T}_3{T}_{34}^{\prime }{}^{D_2^0}P_0$$

(10)

The position representation of point D₃ from the open chain A-D₃ is expressed as follows:

$${}^{D_3^{\prime }}P=T_{4h}^{\prime }{T}_4{T}_{43}^{\prime }{}^{D_3^0}P$$

(11)

The matrices and vectors in Equations (9)–(11) can be found in Appendix A.2. Since the influence of error has been taken into account, ${}^{D_i^{\prime }}P{\ne }^{D_i}P$, the vector of errors ${}_{ }{}^{D_i}{P}_e(i=\mathrm{1,2},3)$ of the mechanism can be expressed as follows [24]:

$${}^{D_1^{\prime }}P_e={|}^{D_1^{\prime }}P{-}^{D_1}P|$$

(12)

$${}^{D_2^{\prime }}P_e={|}^{D_2^{\prime }}P{-}^{D2}P|$$

(13)

$${}^{D_3^{\prime }}P_e={|}^{D_3^{\prime }}P{-}^{D_3}P|$$

(14)

3.3.3. Error Modeling for Closed Loop

In multi-loop mechanisms, there exists a geometric coordination process among the various branches. Therefore, the evaluation of their error differs from that of the branches. Thus, a point position model for the multi-loop mechanism is established based on the geometric coordination relationship to assess the accuracy of the mechanism.

The deployed precision of the core unit of the antenna mechanism is crucial for the performance of the antenna. Therefore, the deployed state is taken as the modeling state, and the mechanism is subjected to tension on the net surface. It is assumed that when the unit is tensioned, the contact point of the hinge is in the direction of the connected rod. Therefore, the virtual rotation angle ${\theta }^{\prime }$ of the hinge gap and the connecting rod is 0 or $\pi$. In the tensioned state, the hole and the shaft are always in contact. Based on the hinge gap and the error in the length of the rod, a geometric error model is established based on geometric coordination, which is shown in Figure 7.

In Figure 7, the points $\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{and} \mathrm{D}$ express the points of the mechanism without gap error and length error. The points ${\mathrm{A}}^{\prime },{\mathrm{B}}^{\prime },{\mathrm{C}}^{\prime },\mathrm{and} {\mathrm{D}}^{\prime }$ express the points of the mechanism with gap error and length error. The distance between point B’ and point C’ is expressed as $l_1^{\prime }$. The distance between point C’ and point D’ is expressed as $l_2^{\prime }$. The distance between point B’ and point D’ is expressed as $l_3^{\prime }$. The distance between point A’ and point D’ is expressed as $l_4^{\prime }$. The distances between the points can be expressed as follows:

$$l_1^{\prime }=l_1+d_1=l_1+{\delta }_1+{\Delta }_1$$

(15)

$$l_2^{\prime }=l_2+d_2=l_2+{\delta }_2+{\Delta }_2$$

(16)

$$l_4^{\prime }=l_4+d_4=l_4+{\delta }_4+{\Delta }_4$$

(17)

$$l_3^{\prime }=l_3+d_3$$

(18)

where ${\delta }_1$ is the gap of hinge gap B. The parameter ${\Delta }_1$ is the length error in rod BC. The parameter ${\delta }_2$ is the gap of hinge gap C. The parameter ${\Delta }_2$ is length error in rod CD. The parameter ${\delta }_4$ is the gap of hinge gap A. The parameter ${\Delta }_4$ is length error in rod AD. $d_1$ is the sum of ${\delta }_1$ and ${\Delta }_1$. $d_2$ is the sum of ${\delta }_2$ and ${\Delta }_2$. $d_4$ is the sum of ${\delta }_4$ and ${\Delta }_4$.

In geometric coordination processing, the distance between point B and point D is changed. The variation in the distance between point B and point D is denoted as $d_3$. Because rod BD is the shared component of the ∆BCD and ∆ABD, based on the law of cosines, the length of rod BD after coordination can be expressed as follows:

$$l_3^{\prime }=\sqrt{l_1^{\prime 2}+l_2^{\prime 2}-2l_1^{\prime }{l}_2^{\prime }cos\alpha }$$

(19)

$$l_3^{\prime }=\sqrt{{l_4^{\prime }}^2+l_6^2-2l_4^{\prime }{l}_6\mathit{cos} \beta }$$

(20)

In ∆A’B’D’, the angle $\gamma$ can be expressed as:

$$\cos \gamma ={\displaystyle \frac{l_6^2+l_3^{\prime 2}-l_4^{\prime 2}}{2l_6{l}_3^{\prime }}}$$

(21)

Combining with Equations (20) and (21) can also be expressed as follows:

$$\cos \gamma ={\displaystyle \frac{l_6^2+l_6^2-2l_4^{\prime }{l}_6\cos \beta }{2l_6{l}_3^{\prime }}}$$

(22)

The value of the $l_6$, $l_4$, and $l_3$ are given. The parameter $l_4^{\prime }$ can be calculated by Equation (17). For the coordinated parameters, parameter d₃ for the distance of point B’ and point D’ can be expressed by parameter $\beta$ using Equation (20):

$$d_3=\pm g\left(l_i,\beta ,{\delta }_4,{\Delta }_4\right), \left(i=\mathrm{3,4},6\right)$$

(23)

Because the pendulum angle of BD is small after the mechanism coordinates, it is assumed that the arc length of ${\mathrm{D}\mathrm{D}}_3$ in the circle with point B as the center and BD as the radius, is equal to the length of ${\mathrm{D}\mathrm{D}}_3$. Similarly, the arc length of ${\mathrm{D}\mathrm{D}}_4$ in the circle with point A as the center and AD as the radius, is equal to the length of ${\mathrm{D}\mathrm{D}}_4$. Then the displacement error in point D after coordination can be expressed as follows:

$${e_{1D}}^2={\left[l_4\cdot \left(\beta -{\displaystyle \frac{\pi }{2}}\right)\right]}^2+d_4^2$$

(24)

$${e_{2D}}^2={\left[l_3\cdot \left({\theta }_5-\gamma \right)\right]}^2+d_3^2$$

(25)

When the value of $\beta$ is set, the value of $\gamma$ can be calculated based on Equation (19). It is found that $e_{1D}$ is equal to $e_{2D}$. The $\beta$ can calculated by numerical simulation. In ∆B’C’D’, the angle $\phi$ which is the angle between rod B’C’ and rod B’D’ can be given as follows:

$$\mathit{cos} \phi ={\displaystyle \frac{l_1^{\prime 2}+l_3^{\prime 2}-l_2^{\prime 2}}{2l_1^{\prime }{l}_3^{\prime }}}$$

(26)

Therefore, in $\mathsf{\Delta }{\mathrm{B}}^{\prime }{\mathrm{C}}^{\prime }{\mathrm{D}}^{\prime }$, the displacement of the point C can be expressed as:

$$e_C=\sqrt{l_1^2+l_1^{\prime 2}-2l_1{l}_1^{\prime }\mathit{cos} \left({\displaystyle \frac{\pi }{2}}-\gamma -\phi \right)}$$

(27)

3.4. Results of Accuracy Analysis

From the previous section, the end error in the deployable unit mechanism is distributed in the x- and y-directions. To facilitate a comparison of the impact of various errors on the end accuracy, the error surfaces for the x-direction, y-direction, and the overall mechanism are output separately below. In order to analyze the influence of hinge gap on the errors in the key points, based on the geometric coordination relationship of the closed-loop system, the error analysis of key points C and D of the mechanism is carried out. The results of point D in the horizontal and vertical directions are shown in Figure 8a,b, and the overall error is shown in Figure 8c.

When the hinge gap and the rod length errors vary within the range [0.1 mm, 0.5 mm], the error range of the mechanism in the x-direction at point D is [0.2 mm, 0.998 mm]. The error range of point D in the y-direction is [−1.792 mm, −0.279 mm]. The total error range of point D is [0.209, 2.05]. As shown in Figure 8, the error at point D of the mechanism follows a linear pattern with respect to the hinge gap and the rod length error. When the hinge gap or the rod length error increases, the x-direction error and the total error also increase accordingly, while the x-direction error changes slowly.

Based on Equation (27), the error in point C can be given in Figure 9.

When the hinge gap and the rod length errors vary within the range of [0.1 mm, 0.5 mm], the x-direction error range of point C is [0.2 mm, 0.995 mm]. The error in point C in the y-direction is between 0.262 mm and 2.792 mm. The total error in point C is between 0.329 mm and 2.964 mm. As shown in Figure 9, the error at point C of the mechanism follows a linear pattern with respect to the hinge gap and the rod length error. When the hinge gap or the rod length error increases, the errors in all directions also increase accordingly.

In the deployed mechanism, the effects of the rod length and the gap error on the error in the key point of the mechanism are equivalent. The error in the different loops has different effects on the key point error. Therefore, the effects of the length error in rods BC, CD, and AD on the key point error in the x- and y-directions are calculated. The total error is also be given in Figure 10, Figure 11 and Figure 12.

In Figure 10, it is shown that the total errors in point C increase monotonically with the errors in the length of rod BC. The error in the x-direction of point C increases linearly with the length BC, while in the y-direction, the change in the error is not significant.

In Figure 11, it is shown that the total errors in point C increase monotonically with the error in the length of rod CD. In the x-direction, the change in the error is not significant. The error in the y-direction of point C increases linearly with the length CD.

The hinge gap of rod AD has a significant impact in the ∆ABD, as shown in Figure 12. The overall error in point D increases monotonically with the length error in rod AD. The error in point D in the x-direction increases with the length error in rod AD. The error in point D in the y-direction has a small change initially and then shows an inverse linear change as the length error in rod AD increases.

The impact of length errors in each link on the point error is presented in

Table 3. The impact of each link error on accuracy.

Link Term	Error			Variation
	X-Direction	Y-Direction	Overall
$\mathrm{Length} \mathrm{error} \mathrm{in} \mathrm{rod} \mathrm{BC} \mathrm{on} \mathrm{point} \mathrm{C} \left(\mathrm{m}\mathrm{m}\right)$	[0.009, 0.959]	[1.213, 1.213]	[1.213, 1.547]	0.334
$\mathrm{Length} \mathrm{error} \mathrm{in} \mathrm{rod} \mathrm{CD} \mathrm{on} \mathrm{point} \mathrm{C} \left(\mathrm{m}\mathrm{m}\right)$	[0.5, 0.498]	[0.724, 1.674]	[0.88, 1.746]	0.866
$\mathrm{Length} \mathrm{error} \mathrm{in} \mathrm{rod} \mathrm{AD} \mathrm{on} \mathrm{point} \mathrm{D} \left(\mathrm{m}\mathrm{m}\right)$	[0.01, 0.958]	[0.01, −1.706]	[0.01, 1.956]	1.955

. It is evident that the length error in link BC has the least effect on the accuracy of point C, while the length error in link CD has a moderate impact. Conversely, the length error in link AD exerts the most significant influence on the accuracy of point D. Hence, the optimization design should focus on analyzing the error at point D, and stringent control over the tolerance range of link AD should be implemented to minimize its effect on the accuracy at point D.

3.5. Simulation Verification

To verify the correctness of the error geometric modeling theory presented in Section 3.3, this section will design a virtual experiment to validate the error at point D. The three-dimensional model is designed based on the link length parameters in Table 1, with the length errors in all links and the hinge gap all set to 0.5 mm, as illustrated in Figure 13.

The 3-D model is saved as a ‘step’ file. Then, it is imported to Workbencth 2022 R1 software. The hinge gap contact state is set. A 5 N/m work surface load is applied to the upper rod and 10 N cable tension forces are applied at the upper and lower connection points of the outer rod as shown in Appendix C. After completing these settings, the position coordinates of the rolling pin at the end of the mechanism are solved to obtain a position coordinate (x₁, y₁). As the control group, the model without errors does not include any errors. The same procedure is repeated to solve for the rolling pin position, obtaining a second position coordinate (x₂, y₂). The simulation error value is calculated based on the modules of the position vector. The results of both roller pin position analyses are systematically presented in

Table 4. Results of both roller pin position analyses.

Item Type	Value
The coordinates of group 1’s position (mm)	Position X1	Position Y1
	750	195.01
The coordinates of group 2’s position (mm)	Position X2	Position Y2
	749.43	204.71
Simulation error value (mm)	1.94509
Theoretical error value (mm)	2.05

.

The percentage error in the experimental results compared with theoretical results is 5.1%. The experimental results demonstrate a remarkable consistency between the theoretical and simulated error values at the end of the deployed unit mechanism. This simulation example validates the accuracy of our theoretical framework but also establishes a reliable foundation for designing space modular deployable antenna systems.

4. Accuracy Synthesis

The main goal of accuracy synthesis is to strike a balance between accuracy and cost, ensuring that the mechanism achieves satisfactory accuracy while minimizing manufacturing costs as much as possible.

4.1. Minimum Cost Mathematical Model

The objective of the minimum cost model is to achieve the minimum processing cost without exceeding a certain error threshold. Let Mc denote the manufacturing and processing cost of the mechanism, where $\left[Mc\right]$ is the allowable cost, and Me represents the error in the six-bar linkage, where $\left[Me\right]$ is the permissible error, where ${\mathrm{e}}_{\mathrm{D}}$.is the positional error in point D. The minimum cost model can be expressed as follows:

$$\left\{\begin{array}{c}\min Mc\\ \mathrm{s}.\mathrm{t}. Me\le \left[Me\right]\end{array}\right.$$

(28)

Assuming the relationship between the manufacturing cost of the mechanism and its tolerance can be calculated using an inverse power exponent model, then the objective function for the minimum cost–accuracy synthesis can be expressed as follows:

$$\min Mc=\sum _{j=1}^N{K}_j{\left(x_j\right)}^{-{\alpha }_j}$$

$$\mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}{\tau }_j\le x_j\le {\sigma }_j\\ min\left|e_D\right|\le \left[\Delta \right]\end{array}\right.$$

(29)

In the formula, ${\alpha }_j$ the processing cost characteristic index, typically ranging from 0.7 to 1; $K_j$ is the manufacturability coefficient; $x_j$ is the error parameter value, then $x=\left[d_1,d_2,d_4\right]$; ${\tau }_j$ is the minimum allowable value for the predefined error parameter, determined by factors such as manufacturing costs and maximum processing capacity; ${\sigma }_j$ is the limit error parameter value; $\left[\Delta \right]$ represents the design accuracy of the mechanism. ${\tau }_j$ and ${\sigma }_j$ serve as the lower and upper bounds.

4.2. Optimal Accuracy Model

Usually, the objective of the optimal accuracy model is to minimize the end error inf the deployable mechanism under the condition that a certain manufacturing and processing cost is not exceeded. Based on the minimum cost model, the optimal accuracy objective function for deployable mechanism accuracy synthesis can be expressed as follows:

$$\min Me=\mathit{inf} \left\{\left|e_D\right|\right\}$$

$$\mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}{\tau }_j\le x_j\le {\sigma }_j\\ {\displaystyle \sum _{j=1}^N}{K}_j{\left(x_j\right)}^{-{\alpha }_j}\le \left[M_c\right]\end{array}\right.$$

(30)

In practical engineering applications, cost information is often difficult to collect, making it challenging to estimate the absolute costs of processing and manufacturing. Moreover, the condition of minimum cost holds no practical significance for the accuracy synthesis of deployable unit mechanisms. Therefore, this study adopts an optimal accuracy model, which eliminates the need for cost information statistics. The objective is to minimize processing costs while ensuring that the lower bound of the optimal accuracy of the deployable unit mechanism meets the required accuracy specifications. The objective function can be expressed as follows [5]:

$$\min M\mathrm{e}=\left|\mathit{inf} \left\{\left|e_D\right|\right\}-\left[\Delta \right]\right|$$

$$s.t.{\tau }_j\le x_j\le {\sigma }_j$$

(31)

Clearly, there are infinitely many sets of x that satisfy Equation (31). As the tolerance band is relaxed, the corresponding manufacturing cost decreases. We aim to select the optimal solution from the infinite set of feasible solutions—that is, under the premise of meeting accuracy requirements, to maximize $x_j$. During the optimization process of accuracy synthesis, an evaluation index I is constructed based on the limit deviation method to assess the relative cost of the obtained solution set. The smaller the value of I, the lower the relative cost. The best accuracy model can be expressed as the following:

$$I=\sum _{j=1}^n{\displaystyle \frac{1}{{\omega }_j}}$$

(32)

In the equation, ${\omega }_j$ is the weight factor. If a certain error has a significant impact on the positional accuracy of point D, the corresponding weight factor should be set to a larger value. The weight factor ${\omega }_j$ can be expressed as follows:

$${\omega }_j=x_j/\sum _{j=1}^n{x}_j$$

(33)

4.3. Optimal Design of Error Parameters

The optimal accuracy synthesis problem is a multidimensional nonlinear constrained optimization problem, making it suitable for applying the numeric iteration to search for the optimal parameter combination. The fitness function for the optimal accuracy model of mechanism synthesis adopts the concept of the weight factor constraints. If the calculated positional error in the points meets the specified accuracy requirements, the fitness function equals the comprehensive error function. The fitness function for mechanism accuracy synthesis can be expressed as follows:

$$fit=\left\{\begin{array}{c}{e}_D\left(x\right), e_D\le \left[\Delta \right]\\ I_{\mathrm{m}}\cdot e_D\left(x\right),\mathrm{otherwise}\end{array}\right.$$

(34)

In the equation: $I_{\mathrm{m}}$ is the cost indicator, $I_{\mathrm{m}}=\sum _{\mathrm{j}=1}^{\mathrm{n}}{\left({\mathsf{\omega }}_{\mathrm{j}}\right)}^{-1}$.

Assuming the design goal for positional accuracy of point D is 0.2 mm, the accuracy design goal considering only component parameter error is set as Δ = 0.2 mm. Based on the optimal accuracy model, the numeric iteration is employed to optimize the allocation of error parameters.

The procedure steps for the synthesis program of the mechanism based on the numeric iteration for the optimal accuracy synthesis model are as follows:

(1).

Initialization: input ${\tau }_j$, $x_j$, ${ \sigma }_j$, β, $\left[\Delta \right]$, ${\mathrm{F}}_m$;

(2).

According to the fitness function for optimal accuracy model, using a numeric method to solve the optimal error $x_{opt}$;

(3).

Calculate the weight factor, ${\omega }_j=x_{optj}/\sum _{\mathrm{j}=1}^{\mathrm{n}}{x}_{optj}$;

(4).

Based on the optimal error $x_{opt}$, calculate the cost indicator, $I_{\mathrm{m}}=\sum _{\mathrm{j}=1}^{\mathrm{n}}{\left({\mathsf{\omega }}_{\mathrm{j}}\right)}^{-1}$;

(5).

Calculate the fitness function, $\mathrm{F}\left(x_{opt}\right)=I_{\mathrm{m}}\cdot {\mathrm{e}}_{\mathrm{D}}\left(x_{opt}\right)$;

(6).

If $\mathrm{F}\left(x_{opt}\right)\le {\mathrm{F}}_m$ then ${\mathrm{F}}_{\mathrm{m}}=\mathrm{F}\left(x_{opt}\right)$, $e_m=e_D\left(x_{opt}\right)$; else, back to step (2); end

(7).

Use $x_{mopt}$ to calculate the positional error function value of the note D, min $e_D\left(x_{opt}\right)$;

(8).

If min $e_D\left(x_{opt}\right)\le \left[\Delta \right]$ then output $x_{mopt}$; else, back to step (2); end

(9).

Terminate the program.

The numerical iterative method was used to solve for the error parameter that minimizes positional error in point D. When $\left[d_1,d_2,d_4\right]=[0.134, 0.134, 0.134]$, the minimum positional error in point D $\min e_D$ = 0.1573 mm, meeting the design requirements. The results of the error parameter optimization allocation are presented in

Table 5. Optimization results of error parameters.

Term	${\mathit{d}}_1/\mathbf{m}\mathbf{m}$	${\mathit{d}}_2/\mathbf{m}\mathbf{m}$	${\mathit{d}}_3/\mathbf{m}\mathbf{m}$	${\mathit{e}}_{\mathit{D}}/\mathbf{m}\mathbf{m}$
Value	0.1340	0.1340	0.1340	0.1573

. The symbolic variables in the text refer to Appendix B.

5. Conclusions

This paper addresses the precision analysis problem of planar multi-loop deployable unit mechanisms and proposes a screw-theory-based sub-chain precision analysis method. This method constructs a virtual screw model of rod length errors and hinge gap errors, which facilitates the expression of relationships between sub-chains and errors. Based on geometric relationships, a multi-loop position error in the nodes model is established, and accuracy surfaces considering rod length errors and hinge gap are output using MATLAB. By outputting the relationship curves of single-rod errors relative to point errors, the linearized influence law of individual rods on precision is further elucidated. Simulation results demonstrate the reliability of the error modeling theory. A cost-effective precision model is established, and based on the numerical iterative method, the minimum point error is solved to obtain the optimal solution for error parameters.

However, this geometric modeling approach has certain limitations, such as the extremely complex error-solving process for spatial multi-loop models. In the future, attempts will be made to establish a precision model by analyzing the error transmission laws between loops.